Nonlinear Wave Motion

波浪运动本质上是基本的：光波影响人们看的方式，声波让人们听到，水波对船只的运动和潮汐能量至关重要。该项目专注于非线性波传播，即研究大幅度或高功率信号，旨在提高科学家对大幅度波及其相关的波能传递动力学的洞察和理解的能力。研究结果将有助于非线性波动的应用，包括海啸、流浪波或畸形波、高功率激光，以及压力波的大变化传播，如冲击波和冲击波。对于线性波或小振幅波，有一个坚实的理论，在许多情况下，可以找到精确的解。对于非线性波动，现有的理论还不是很成熟，已知精确解的非线性波动方程也相对较少。然而，所谓的逆散射变换方法被用来求解和了解基本方程的重要性质，并导致了一类具有物理意义的非线性波动方程的解。这种方法可以用来寻找所谓的孤子解，即稳定的局域波，它具有特殊的相互作用性质，并已被广泛应用于水波、电磁波、晶格动力学、磁波和弹性。研究人员将考虑将该方法推广到新的一类非线性方程，如分数阶非线性波动方程及其孤子解。分数阶方程本身有许多应用，包括对非晶态材料的研究。此外，该方法还将用于研究一类多维孤子的性质，即在所有方向上消失的块状波，它也与量子力学和光学中出现的方程相关。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Wave motion is fundamental in nature: Light waves impact the way people see, sound waves allow people to hear, and water waves are critical to the motion of ships and tidal energy. This project is focused on nonlinear wave propagation, that is the study of large amplitude or high-power signals, and aims to enhance the ability of scientists to develop insight and understanding of large amplitude waves and of the associated dynamics of wave energy transfer. The results of the investigation will benefit applications of nonlinear wave motion, which include tsunamis, rogue or freak waves, high power lasers, and the propagation of large changes in pressure waves such as those that occur in shock waves and blast waves.For linear waves or small amplitude waves a substantial theory is available and, in many situations, exact solutions can be found. For nonlinear waves the available theory is less developed and there are relatively few nonlinear wave equations for which exact solutions are known. However, the so-called Inverse Scattering Transform method has been used to find solutions and understand important properties of the underlying equations and has led to a wide class of solutions of physically significant nonlinear wave equations. This method can be used to find so-called solitons solutions, stable localized waves which have special interaction properties and have been found to apply widely, for example in water waves, electromagnetic waves, lattice dynamics, magnetic waves, and elasticity. The investigator will consider extensions of this method to new classes of nonlinear equations, such as fractional nonlinear wave equations and their soliton solutions. Fractional equations themselves have numerous applications, including to the study of amorphous materials. Additionally, the method will be applied to study the properties of a class of multidimensional solitons, that is lump-type waves that vanish in all directions, which are relevant also for equations that arise in quantum mechanics and optics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.